Question

Suppose John is a high school statistics teacher who believes that eating candy improves focus, which leads to higher test scores. Immediately after giving the most recent test, he surveyed each of the 35 students in his class and asked them how many individual candies they ate that week. He then matched each student's test grade with his or her survey response. After compiling the data, he used the number of candies eaten to predict each student's test score. He found the least-squares regression line to be ?̂=2.8?+86y^=2.8x+86. He also calculated that the value of ?r, the correlation coefficient, was 0.54.

Which of the choices identifies the correct value of the coefficient of determination, ?2R2, and gives a correct interpretation of its meaning?

?2=0.54R2=0.54, meaning 54% of the total variation in test scores can be explained by the least‑squares regression line.

?2=0.2916R2=0.2916, meaning 29.16% of the total variation in test scores can be explained by the least‑squares regression line.

?2=0.54R2=0.54, meaning 54% of the total variation in the number of candies eaten can be explained by the least‑squares regression line.

?2=0.7348R2=0.7348, meaning 73.48% of the total variation in test scores can be explained by the least‑squares regression line.

?2=0.2916R2=0.2916, meaning 29.16% of the total variation in the number of candies eaten can be explained by the least‑squares regression line.

Answer 1

Solution:

Suppose John is a high school statistics teacher who believes that eating candy improves focus, which leads to higher test scores.

Thus y = dependent variable = test scores and x = independent variable = the number of candies eaten

We have the least-squares regression line :

the correlation coefficient = r =  0.54

Which of the choices identifies the correct value of the coefficient of determination, R², and gives a correct interpretation of its meaning?

Since r = 0.54

then

coefficient of determination, R², gives amount of variation explained in dependent variable y, by least-square regression line.

Thus from given options:

First Option says R2 = 0.54 which is incorrect or not equal to 0.2916

Second option : R²=0.2916, meaning 29.16% of the total variation in test scores can be explained by the least‑squares regression line.

Since our dependent variable is Test scores and we obtained ,

and second option says: 29.16% of the total variation in test scores can be explained by the least‑squares regression line.

Thus second option is correct.

Options 3 and 4 does not equal to   R²=0.2916,

and option 5 is equal to   R²=0.2916, but it says the total variation in the number of candies eaten can be explained by the least‑squares regression line,but the number of candies eaten is not dependent variable.

Thus option 1, 3 , 4 and 5 are incorrect.